Let f(x) = x² + 9x / (3x−1)(x² + 3)

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

1. B1 State or imply the form A/(3x-1) + (Bx+C)/(x²+3)
2. M1 Use a correct method for finding a constant
3. A1 Obtain one of A = 1, B = 0 and C = 3 from correct working A maximum of M1 A1 is available after B0.
4. A1 Obtain a second value from correct working
5. A1 Obtain the third value from correct working
6. B1 FT Integrate and obtain term (1/3)ln(3x-1) OE e.g. (1/3)ln(x-1/3). The FT is on the value of A.
7. M1 Obtain term of the form k tan⁻¹(x/√3)
8. A1 FT Obtain term √3tan⁻¹(x/√3) OE. The FT is on the value of C.
9. M1 Substitute correct limits in an integral of the form aln(3x-1)+k tan⁻¹(x/√3) where ak ≠ 0, and evaluate trigonometry Must be subtracted the right way round. ((1/3)ln8 - (1/3)ln2 + √3tan⁻¹(√3) - √3tan⁻¹(1/√3)) Angles should be in radians. Condone angles as decimals.
10. A1 Obtain answer (2/3)ln2 + π/√3 from correct working in part 8(b) Or exact 2-term equivalent e.g. (1/3)ln4 + π/(2√3) ISW.